论文标题
均值旋转眼镜的优化
Optimization of Mean-field Spin Glasses
论文作者
论文摘要
平均场自旋眼镜是高维产品空间上随机能量功能（哈密顿量）的家族。在本文中，我们考虑了混合$ p $ spin型号的情况，即hamiltonians $ h_n：σ_n\ to {\ mathbb r} $上的hamming hamming hamming hamming hypercube $σ_n= \ {\ pm {\ pm 1 \ pm}^n $，这些属性是由该属性定义的$ \ \ boldsym \} _ {{{\ boldsymbolσ} \ inσ_n} $是一个中心的高斯过程，具有协方差$ {\ mathbb e} \ {h_n（{\ boldsymbolσ} _1）H_n（{ $ \ langle {\boldsymbolσ} _1，{\boldsymbolσ} _2 \ rangle $。最佳$ \ max _ {{{\ boldsymbolσ} \ inσ_n} h_n（{\ boldsymbolσ}）$的渐近价值是根据talagrand和更一般性的panchennko在Parisi公式中被称为Parisi Formula的各种原则的特征。 Supervel套装的结构非常丰富，并且已经由许多作者研究。在这里，我们询问是否可以在多项式时间内计算几乎最佳的配置$ {\boldsymbolσ} $。我们开发了传递算法的消息，该算法的复杂性均与评估$ H_N $的梯度的复杂性相同，并表征其实现的典型能量值。当$ p $ -spin模型$ h_n $满足某个无重叠的差距假设时，对于任何$ \ varepsilon> 0 $，算法输出$ {\ boldsymbolσ} \inς_n$，使得$ h_n（ （1- \ varepsilon）\ max _ {{\boldsymbolσ}'} h_n（{\boldsymbolσ}'）$，具有很高的概率。迭代次数以$ n $为界，并取决于$ \ varepsilon $。更一般而言，无论是否存在无限制差距假设，所获得的能量都是由扩展的变异原理给出的，该原理概括了巴黎公式。
Mean-field spin glasses are families of random energy functions (Hamiltonians) on high-dimensional product spaces. In this paper we consider the case of Ising mixed $p$-spin models, namely Hamiltonians $H_N:Σ_N\to {\mathbb R}$ on the Hamming hypercube $Σ_N = \{\pm 1\}^N$, which are defined by the property that $\{H_N({\boldsymbol σ})\}_{{\boldsymbol σ}\in Σ_N}$ is a centered Gaussian process with covariance ${\mathbb E}\{H_N({\boldsymbol σ}_1)H_N({\boldsymbol σ}_2)\}$ depending only on the scalar product $\langle {\boldsymbol σ}_1,{\boldsymbol σ}_2\rangle$. The asymptotic value of the optimum $\max_{{\boldsymbol σ}\in Σ_N}H_N({\boldsymbol σ})$ was characterized in terms of a variational principle known as the Parisi formula, first proved by Talagrand and, in a more general setting, by Panchenko. The structure of superlevel sets is extremely rich and has been studied by a number of authors. Here we ask whether a near optimal configuration ${\boldsymbol σ}$ can be computed in polynomial time. We develop a message passing algorithm whose complexity per-iteration is of the same order as the complexity of evaluating the gradient of $H_N$, and characterize the typical energy value it achieves. When the $p$-spin model $H_N$ satisfies a certain no-overlap gap assumption, for any $\varepsilon>0$, the algorithm outputs ${\boldsymbol σ}\inΣ_N$ such that $H_N({\boldsymbol σ})\ge (1-\varepsilon)\max_{{\boldsymbol σ}'} H_N({\boldsymbol σ}')$, with high probability. The number of iterations is bounded in $N$ and depends uniquely on $\varepsilon$. More generally, regardless of whether the no-overlap gap assumption holds, the energy achieved is given by an extended variational principle, which generalizes the Parisi formula.
